Question

Lim
T-44
√x+5-7
x - 44

Answer 1

`\displaystyle \lim_(x\to 44)~\cfrac{√(x+5)-7}{x-44}\hspace{5em}\stackrel{\textit{L'Hopital's rule}}{\lim_(x\to 44)~\cfrac{ ~~ (d)/(dx)[√(x+5)-7] ~~ }{(d)/(dx)[x-44]}} \\\\[-0.35em] ~\dotfill`

`\cfrac{d}{dx}[√(x+5)-7]\implies \cfrac{1}{2}(x+5)^{-(1)/(2)}(1)\implies \cfrac{1}{2√(x+5)} \\\\\\ \cfrac{d}{dx}[x-44]\implies 1 \\\\[-0.35em] ~\dotfill\\\\ \displaystyle \lim_(x\to 44)~\cfrac{√(x+5)-7}{x-44}\implies \lim_(x\to 44)~\cfrac{ ~~ (1)/(2√(x+5)) ~~ }{1}\implies \lim_(x\to 44)~\cfrac{1}{2√(x+5)}\implies \cfrac{1}{2√(44+5)} \\\\\\ \cfrac{1}{2√(49)}\implies \cfrac{1}{2(7)}\implies \cfrac{1}{14}`